Functional Dependence between the Hamiltonian and the Modular Operator Associated with a Faithful Invariant State of a $W^{\ast }$-Dynamical System

Abstract

Let $H$ and $\Delta$ be the hamiltonian, resp. the modular operator associated with an invariant faithful normal state $\omega$ of a $W^{\ast }$-dynamical system $(A,\alpha )$. Then $\Delta =f(h)$ for some decreasing function $f$ if and only if (roughly speaking) $\omega$ is 2-passive with respect to $\alpha$. It follows that under certain conditions a 3-passive state is an equilibrium (i.e. KMS) state.